Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 7.6, Problem 15E
Interpretation Introduction

Interpretation:

To show that frequency of small oscillations of amplitude a1 is ω1116a2

To show that formula for ω is consistent with the exact result T = 2π(1+116a2+O(a4))

Concept Introduction:

The system equation for linear oscillator is x¨  + x = 0. If the system is perturbed by small perturbation constant, the system equation becomes x¨ + x + εh(x,x˙)= 0

Here, 0<ε1, and h(x,x˙) is smooth function.

This system is known as weakly nonlinear oscillator.

The polar transformation of Cartesian co-ordinates is

x = rcosθy = rsinθ

The averaged equations for r and rϕ are

r=10h(θ)sinθdθh sinθ

rϕ=10h(θ)cosθdθh cosθ

Where h(θ)=h(x,x˙)=h(r cosθ,r sinθ)

The value of r0 is calculated as:

r0=(x(0))2+(x˙(0))2

The value of ϕ0 is calculated by:

ϕ0=tan(x˙(0)x(0))

The exact formula for time period is T = 2π(1+116a2+O(a4))

The Taylor’s series expansion for (1x)1 is

(1- x)-1=1+x+x2+x4+......

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